This coincides with \(a_t\)  if \(x_t=1\)  and with \(b_t\)  if the last trader
was a seller (\(x_t=0\) ). The price satisfies the martingale
propertyThe price \(p_{t+1}=\mathbb {E}[Y|x_{\le t},X_{t+1}]\) 
at time \(t\)  is a random variable, because it depends on the
realisation \(X_{t+1}\)  of the next trade.
\(\mathbb {E}[p_{t+1}|x_{\le t}]=\mathbb {E}[\mathbb {E}[Y|x_{\le t},X_{t+1}]]\) 
is the expected value of this random variable on \(X_{t+1}\) . By the
property of expectations, this equals \(\mathbb {E}[Y|x_{\le t}]=p_t\) .
\(\mathbb {E}[p_{t+1}|x_{\le t}]=p_t\)  [1]}.
